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arXiv:2608.29333

Zero-Energy Problems for Supersymmetric Hamiltonians on a Chain Are $\mathrm{QMA}_1$-Complete

Yibin Wang

quant-phSubmitted Aug 29, 2026• Announced Sep 1, 2026• 0 citations• Revised Oct 2, 2026 (v3)
v1 · Aug 29, 2026v2 · Sep 1, 2026v3 · Oct 2, 2026
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Abstract

We determine the computational complexity of deciding whether a supersymmetric quantum system on a one-dimensional chain has a zero-energy ground state. We prove $\mathrm{QMA}_1$-completeness for two input models, formulated in terms of Hermitian and nilpotent supercharges, respectively. In the Hermitian model, both the supercharge terms and the Hamiltonian have bounded range on the chain; in the nilpotent model, this geometric restriction applies only to the Hamiltonian. The hardness reduction from quantum satisfiability on a chain preserves exact zero energy and gives an inverse-polynomial ground-energy lower bound for negative instances. A common verification method establishes membership in $\mathrm{QMA}_1$ while allowing cancellations between supercharge terms.

Comments
39 pages. Corrected and simplified the proofs in Appendix B; added explanatory figures

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